My data is given by the Fourier transform of the function, where the points are distributed in a ball with uniformly distributed distances and uniformly distributed spherical angles (not Gaussian angles).
So the grid in Fourier space is obviously non-uniform (uniform spherical angles imply non-uniform distribution on the sphere).
I need to reconstruct the function from such data. I don't care yet about effectiveness of the algorithm but I want to know if it is in principal possible to reconstruct it from such data. I know that reconstruction is very sensitive to the grid in Fourier space.
p.s. I know that in 2D for example, the uniform polar-coordinates grid is ok.
p.p.s I tried to do the inversion by discretizing the Fourier integral in 3D -- so it will be the summ of all points in the ball multipilied by respective exponents and multiplied by discretized jacobian (in spherical coordinates). The pictures I get are unsatisfactory.
On this picture it should be a small square in the middle (a slide of a square in 3D).
The answer is yes. Sorry to taking your time with questions. Naive discretization of the Fourier integral gives already meaningful results.
Reconstruction of a slice with square potential(with post-smoothing)
Reconstruction of a slice with round potential (no postsmoothing)
